Quantum graph models of physical systems

The concept of quantum graphs was proposed in early years of quantum theory [1] but then forgotten for half a century. Its revival is connected with the progress in solid-state physics, where it serves as models of nanostructures, see e.g. [2,3], or metamaterials [4], however, quantum graphs also provide a versatile tool to study various effects, for instance, quantum chaos [5]. There are several monographs describing state of the arts, e.g. [6,7].

The proposed thesis subject aims at some open questions in this area, in particular,

• Modelling material textures through dense graph limits. Such a result exists for square lattices [8], the aim is to extend it to other types, both those for which the underlying problem for discrete Laplacian is solved [9] and those for which it is not.

• Transport in lattices due to infinitely extended variation of the geometry or vertex coupling. A prototypic example is solved in [10], the aim is to deal with a broader class of lattices and perturbations.

• Modelling the anomalous Hall effect: an attempt to employ quantum graphs to this aim was made in [11], however, it used an assumption which cannot be justified from the first principles. The aim is to construct such a model using the vertex coupling violating the time-reversal invariance [12].

• Analysis of quantum graphs depending on parameters, both in geometry and the vertex coupling, focusing both on time dependence [14] and Berry phase [15].

[1] L. Pauling: The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys. 4 (1936), 673-677.

[2] R.A. Webb et al.: Observation of h/e Aharonov-Bohm oscillations in normal-metal rings, Phys. Rev. Lett. 54 (1985), 2696-2699.

[3] A. Führer et al.: Energy spectra of quantum rings, Nature 413 (2001), 822-825.

[4] T. Lawrie, G. Tanner, D. Chronopoulos: A quantum graph approach to metamaterial design, Sci. Rep. 12 (2022), 18005

[5] T. Kottos, U. Smilansky: Quantum chaos on graphs, Phys. Rev. Lett. 79 (1997), 4794-4797.

[6] G. Berkolaiko, P. Kuchmunt: Introduction to Quantum Graphs, Amer. Math. Soc., Providence, R.I., 2013.

[7] P. Kurasov: Spectral Geometry of Graphs, Birkhäuser, Berlin 2024.

[8] P. Exner, S. Nakamura, Y. Tadano: Continuum limit of the lattice quantum graph Hamiltonian, Lett. Math. Phys. 112 (2022), 83

[9] K.Mikami, S. Nakamura, Y. Tadano: Continuum limit for Laplace and elliptic operatorson lattices, Pure App. Anal. 6 (2024), 765-788.

[10] M. Baradaran, P. Exner, A. Khrabustovskyi: Quantum lattice transport along an infinitely extended perturbation, J. Phys. A.: Math. Theor. 58 (2025), 145203

[11] P. Středa, K. Výborný: Anomalous Hall conductivity and quantum friction, Phys. Rev. B107 (2023), 014425

[12] P. Exner, M. Tater: Quantum graphs: self-adjoint, and yet exhibiting a nontrivial PT-symmetry, Phys. Lett. A416 (2021), 127669.

[14] D.U. Matrasulov, J.R. Yusupov, K.K. Sabirov, Z.A. Sobirov: Time-dependent quantum graph, Nanosystems: Physics, Chemistry, Mathematics 6 (2015), 173-181.

[15] P. Exner, H. Grosse: Some properties of the one-dimensional generalized point interactions (a torso), math-ph/9910029